Matrix

Eigenvectors: find eigenvalues, then solve

A−λI=0A−λI=0

Lagrange

ddt(∂L∂x˙)−∂L∂x=0\frac{d}{dt}\left(\frac{\partial{\mathcal{L}}}{\partial{\dot{x}}}\right)−\frac{\partial{\mathcal{L}}}{\partial{x}}=0

Structures

Euler Bernoulli

Euler-Bernoulli EquationEuler-Bernoulli Equation · ^Euler-Bernoulli-equation · Embedded block

EId2ydx2=F(x)EI\frac{d^2y}{dx^2}=F(x)

FEM

Bending Matrix

FEMEmbedded note
K=EIL3[126L−126L6L4L2−6L2L−12−6L12−6L6L2L2−6L4L2]K = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}

K=[1−1−11]K= \begin{bmatrix} 1 & -1 \\ -1 & 1 \\ \end{bmatrix}

Ku=FKu=F

Bending MatrixFEM · ^element-stifness-matrix · Embedded block
K=EIL3[126L−126L6L4L2−6L2L−12−6L12−6L6L2L2−6L4L2]K = \frac{EI}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}

Axial Stress Matrix

K=[1−1−11]K= \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}


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